"Question: Can I put my jars into cold water and then bring them to a boil and start the timing at that point?"
You sure Can can that way!! I have done it before. You start timing when the water reaches a rolling boil.
Here's the deal, the jars can stand some temperature changes but Will crack when you go from cold to hot OR hot to cold. My expertise is in hot bath canning---STILL haven't used my pressure canner, but DH is threatening to cut something off if I don't use it this year!!

I put the rack in FIRST, and then add the jars. I add HOT food to HOT jars and making them wait until you have filled the rack will let them cool some. SOME products, like jelly will keep their heat pretty well.
Most of the time it is a good idea to have an extra pot, or a small crock pot, with boiling hot water in it. You need the piece that has a magnet so that you can put the lids in and soften up the rubber RIGHT BEFORE you put it on the jar, then you hand screw on the lid as tight as possible, and then
as soon as you can you put it in the rolling boiling hot water bath, put on the lid, and start the timer. Better to go longer than shorter.
I remove mine to my stove top and then to a large wooden cutting board to be sure that the colder counter doesn't crack a jar. When you hear the pop, it means that a jar has completed it's sealing. Always test the lid center. If it is not solid you can press down and it will pop back. THAT means that it is NOT sealed. Not a problem! Put it back in and cook it long enough to finish the seal. With practice you will get the hang of it!

When you do a batch, the last set of jars seems like baking the last cookies from a batch and you are DONE with it!!

I often check the jars, turn off the burner and leave them sit in the water overnight, then take them out in the morning. Doesn't hurt a thing.
Lost ONE jar out of hundreds in the last 5 years bc of a bad seal.
I used "can" 5x
You will find Other uses for the funnels, and I have 2 of them. I use some of my quart jars to stores other things, and my coffee beans have lids that I bought specifically for NON SEALED storage. It is very easy to pour ANYTHING into canning jars with these funnels and I buy 2lb packages of coffee beans. They are decorative and give you other ways to use your jars, BUT, you cannot use them to Can with, in case you didn't know.
https://www.walmart.com/ip/The-Pioneer-Woman-Spring-Mason-Storage-Jar-Lids-Set-of-4/781737612
https://www.youtube.com/watch?reload=9&v=y2he3gF5uSM